Download - CS 460 Spring 2011
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CS 460 Spring 2011
Lecture 3Heuristic Search / Local Search
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Review• Problem-solving agent design and implementation
– PEAS• Vacuum cleaner world• Problem solving as search
– State space (abstraction), problem statement, goal, solution• Search: tree search, graph search, • Search strategies
– Uninformed vs informed– Graph vs. Tree: keep track of nodes already explored
• Uninformed– BFS, Uniform-cost, DFS + variant
• Evaluation of strategies– Completeness, time complexity, space complexity, optimality
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Review: Tree search
A search strategy is defined by picking the order of node expansion
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Graph Search
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Informed search algorithms
Chapter 3 & 4
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Material
• 3.5.1, 3.5.2, 3.6, 4.1••••
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Outline• Best-first search• Greedy best-first search• A* search• Heuristics• Local search algorithms• Hill-climbing search• Simulated annealing search• Local beam search• Genetic algorithms
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•Informed Search• Use additional information about problem to guide
the search• E.g., in Romanian travel problem, estimated
distance from a node to destination as the crow flies• “heuristic”
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Best-first search• Idea: use an evaluation function f(n) for each node• estimate of "desirability"
–– Expand most desirable unexpanded node
• Implementation:– Order the nodes in fringe (frontier in ed 3) in decreasing order of desirability– Compare with uniform cost strategy
• Special cases:– greedy best-first search– A* search–
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Romania with step costs in km
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Greedy best-first search
• Evaluation function f(n) = h(n) (heuristic)• = estimate of cost from n to goal•• e.g., hSLD(n) = straight-line distance from n to
Bucharest•• Greedy best-first search expands the node
that appears to be closest to goal•
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Greedy BeFS vs Uniform Cost
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Properties of greedy best-first search
• Complete? No – can get stuck in loops, e.g., Iasi Neamt Iasi Neamt
• (from Iasi to Fagaras, using tree search)
Time? O(bm), but a good heuristic can give dramatic improvement
•• Space? O(bm) -- keeps all nodes in memory•• Optimal? No• (shorter to go to Bucharest through Rimnicu VilceaPitesti)
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A* search
Idea: avoid expanding paths that are already expensive•• Evaluation function f(n) = g(n) + h(n)•• g(n) = cost so far to reach n• h(n) = estimated cost from n to goal• f(n) = estimated total cost of path through n to goal• Compare with uniform cost search• Uses ony g(n)
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A* search example
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A* search example
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A* search example
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A* search example
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A* search example
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A* search example
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Admissible heuristics• A heuristic h(n) is admissible if for every node n,• h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n.
• An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic
• (SLD uses triangle inequality)• By trying to achieve a underestimate of the cost, you get close to achieving
optimal cost••• Example: hSLD(n) (never overestimates the actual road distance)•• Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal•
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Optimality of A* (proof)Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.•
• f(G2) = g(G2) since h(G2) = 0 • g(G2) > g(G) since G2 is suboptimal • f(G) = g(G) since h(G) = 0 • f(G2) > f(G) from above
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Optimality of A* (proof)Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.•
• f(G2) > f(G) from above • h(n) ≤ h^*(n) since h is admissible• g(n) + h(n) ≤ g(n) + h*(n) • f(n) ≤ f(G)•• Hence f(G2) > f(n), and A* will never select G2 for expansion
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Consistent heuristicsA heuristic is consistent if for every node n, every successor n' of n generated by any action a, •
h(n) ≤ c(n,a,n') + h(n')
If h is consistent, we have•f(n') = g(n') + h(n') = g(n) + c(n,a,n') + h(n') ≥ g(n) + h(n) = f(n)
• i.e., f(n) is non-decreasing along any path.• (start with lowest optimistic estimate, then ‘sacrifice’ some optimality at every step taken)•• Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal•
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Optimality of A*
A* expands nodes in order of increasing f value•
• Gradually adds "f-contours" of nodes • Contour i has all nodes with f=fi, where fi < fi+1
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Properties of A$^*$
Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) )•• Time? Exponential•• Space? Keeps all nodes in memory•• Optimal? Yes• Heuristics are needed to make time and space
manageable•
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Admissible heuristicsE.g., for the 8-puzzle:
• h1(n) = number of misplaced tiles• h2(n) = total Manhattan distance• (i.e., no. of squares from desired location of each tile)
• h1(S) = ? • h2(S) = ? •
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Admissible heuristicsE.g., for the 8-puzzle:
• h1(n) = number of misplaced tiles• h2(n) = total Manhattan distance• (i.e., no. of squares from desired location of each tile)
• h1(S) = ? 8• h2(S) = ? 3+1+2+2+2+3+3+2 = 18
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Dominance• If h2(n) ≥ h1(n) for all n (both admissible)• then h2 dominates h1 • h2 is better for search•
Typical search costs (average number of nodes expanded):•
• d=12 IDS = 3,644,035 nodesA*(h1) = 227 nodes A*(h2) = 73 nodes
• d=24 IDS = too many nodesA*(h1) = 39,135 nodes A*(h2) = 1,641 nodes
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Relaxed problems
A problem with fewer restrictions on the actions is called a relaxed problem•• The cost of an optimal solution to a relaxed problem is
an admissible heuristic for the original problem•• If the rules of the 8-puzzle are relaxed so that a tile can
move anywhere, then h1(n) gives the shortest solution•• If the rules are relaxed so that a tile can move to any
adjacent square, then h2(n) gives the shortest solution•
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Local search algorithms
In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution•
• State space = set of "complete" configurations• Find configuration satisfying constraints, e.g., n-
queens
• In such cases, we can use local search algorithms• keep a single "current" state, try to improve it• Thus also address memory limitations of heuristic searches
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Example: n-queens
Put n queens on an n × n board with no two queens on the same row, column, or diagonal•
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Hill-climbing search
"Like climbing Everest in thick fog with amnesia"•
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Hill-climbing search
Problem: depending on initial state, can get stuck in local maxima•
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Hill-climbing search: 8-queens problem
• h = number of pairs of queens that are attacking each other, either directly or indirectly
• h = 17 for the above state•
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8 queens
• succ: move a single queen to another square in same column• Each state has 8 x 7 = 56 successor states• Number in each square represents total number of attack pairs
for the state resulting when queen in that column is moved to that square
• Choose randomly among ‘best’ successors if there is more than one
• Greedy local search– Doesn’t always get to the global optimum, only local– In the example, we get to h=1 in 5 moves but can’t do better – Local maxima, ridges, plateaux– Use sideways moves, have to cutoff to avoid infinite loops
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Hill-climbing search: 8-queens problem
A local minimum with h = 1•
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Hill climbing• Steepest ascent HC• Stochastic HC
– First choice HC• Random restart HC• Hill climbing algorithms are usually incomplete
– Guaranteed to be incomplete if no downhill moves• But, with effective variants, HC can converge quite fast to a
solution• 8^8 states for 8 queens
– Can solve in about 22 steps on average• Can find solution in under a minute for 3M queens
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Simulated annealing search
Idea: escape local maxima by allowing some "bad" moves but gradually decrease their frequency•
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Properties of simulated annealing search
One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1•
Widely used in VLSI layout, airline scheduling, etc•
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Local beam searchKeep track of k states rather than just one•
Start with k randomly generated states•
At each iteration, all the successors of all k states are generated•
If any one is a goal state, stop; else select the k best successors from the complete list and repeat.•
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Genetic algorithmsA successor state is generated by combining two parent states•
Start with k randomly generated states (population)•
A state is represented as a string over a finite alphabet (often a string of 0s and 1s)•
Evaluation function (fitness function). Higher values for better states.•
Produce the next generation of states by selection, crossover, and mutation•
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Genetic algorithms
Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28)•• 24/(24+23+20+11) = 31%•• 23/(24+23+20+11) = 29% etc•
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Genetic algorithms